EXAMPLE 5 Use inverse properties Simplify the expression
- Slides: 5
EXAMPLE 5 Use inverse properties Simplify the expression. b. log 5 25 x a. 10 log 4 SOLUTION a. 10 log 4 = 4 b. log 5 25 x b x 2 5 log b x =x = log 5 ( ) Express 25 as a power with base 5. = log 5 52 x Power of a power property = 2 x logb bx = x
EXAMPLE 6 Find inverse functions Find the inverse of the function. a. y=6 x b. y = ln (x + 3) SOLUTION a. From the definition of logarithm, the inverse of x y = 6 is y = log 6 x. b. y = ln (x + 3) x = ln (y + 3) ex = (y + 3) ex – 3 = y ANSWER Write original function. Switch x and y. Write in exponential form. Solve for y. The inverse of y = ln (x + 3) is y = ex – 3.
GUIDED PRACTICE for Examples 5 and 6 Simplify the expression. 10. 8 log 8 x SOLUTION 8 log 8 x = x b log b b =x 11. log 7 7– 3 x SOLUTION log 7 7– 3 x = – 3 x log a ax = x
GUIDED PRACTICE for Examples 5 and 6 Simplify the expression. 12. log 2 64 x SOLUTION log 2 64 x = log 2 (26) x Express 64 as a power with base 2. = log 2 26 x Power of a power property = 6 x logb bx = x 13. eln 20 SOLUTION eln 20 = e log e 20 = 20 e log e x = x
GUIDED PRACTICE for Examples 5 and 6 14. Find the inverse of y = 4 x SOLUTION From the definition of logarithm, the inverse of y = 6 is y = log 4 x. 15. Find the inverse of y = ln (x – 5). SOLUTION y = ln (x – 5) x = ln (y – 5) ex = (y – 5) ex + 5 = y ANSWER Write original function. Switch x and y. Write in exponential form. Solve for y. The inverse of y = ln (x – 5) is y = ex + 5.
